Harmonic Oscillator Theory Research Articles

An approach to calculate Franck-Condon factors involving anharmonic potentials using harmonic oscillator bases

In this contribution a discrete variable representation (DVR) approach to calculate Franck-Condon factors (FCFs) is presented. The method is illustrated using a harmonic oscillator basis. The advantage of this approach is that it is possible to calculate FCFs associated with general potentials based on the knowledge of the harmonic oscillator theory. First the coordinate and momentum representations are introduced. Based on this concept, the DVR is proposed, which allows the Franck-Condon factors to be calculated in a simple form. This method takes advantage of the momentum as generator of translations to include the displacement of the potentials. The calculation of the FCFs is exemplified for transitions involving the ground state X1Σg+ and the excited state a 1Π g of the diatomic molecule 14N2 both modelled with Morse potentials.

Numerical study on role of foot of a flue organ pipe: relative phases in oscillations among pipe, foot and jet

Recent numerical studies showed that the foot of flue organ pipes closed at one end acts as a Helmholtz resonator and influences the acoustic oscillation in the pipe. Namely, the relative phase of oscillations between the pipe and foot changes depending on the resonance frequency of the foot. This fact can be explained by the theory of forced harmonic oscillators, in which the pipe works as a force driving the foot. However, in the damping-controlled regime, in which the resonance frequency of the foot, fH, is nearly equal to the frequency of acoustic oscillation, fa, the amplitude of the acoustic oscillation is smaller than those in the mass-controlled regime fH < fa and the stiffness-controlled regime fH > fa, even though the pressure oscillation in the foot lags behind that in the pipe by π/2. In this paper, to investigate this problem and to explore the role of the foot in detail, using a three-dimensional model of a closed flue organ pipe, we numerically investigate the relative phases among the pressure oscillation in the pipe, that in the foot and the jet velocity. We conclude that total balance among them is crucial for sustaining acoustic and fluid oscillations in the sound generation process. The oscillating jet drives the pipe with appropriate timing to sustain the acoustic oscillation for the stiffness-controlled and mass-controlled regimes. In contrast, the jet inefficiently drives the pipe with inappropriate timing for the damping-controlled regime.

Kekulé’s Oscillating D3h Cyclohexatriene Structure of Benzene

AbstractKekulé postulated that neighbouring carbon atoms in benzene undergo incessant collision with each other, thereby leading to the interchange of double and single bonds, which amounts to an oscillation between two cyclohexatriene structures in dynamic equilibrium. It has been claimed that Kekulé arrived at a fully symmetric D6h structure of benzene and that the oscillation hypothesis should not be attributed to him. However, Clausius’ collision theory, which was known at the time, implies that, when the absolute temperature approaches zero, the collision frequency will tend toward zero too, i.e. collisions will stop, and a static, D3h cyclohexatriene obtains. The classical collision theory did not allow Kekulé to construct the desired D6h structure as the energy minimum. The theory of harmonic oscillators would have allowed it, but that was not attempted at Kekulé’s time.

The role of noise in the early universe

In this paper, we consider a quantum mechanical system to model the effect of quantum fields on the evolution of the early universe. The system consists of an inverted oscillator bilinearly coupled to a set of harmonic oscillators. We point out that the role of noise may be crucial in the dynamics of the oscillator, which is analyzed using the theory of harmonic oscillators with random frequency. Using this analogy, we argue that due to the fluctuations around its mean value, a positive vacuum energy density would not produce an exponentially expanding but an oscillating universe, in the same fashion that an inverted pendulum is stabilized by random oscillations of the suspension point (stochastic Kapitza pendulum). The results emphasize the relevance of noise in the evolution of the scale factor.

Heavy Particle Impact Vibrational Excitation and Dissociation Processes in CO2.

A heavy particle impact vibrational excitation and dissociation model for CO2 is presented. This state-to-state model is based on the forced harmonic oscillator (FHO) theory, which is more accurate than current state-of-the-art kinetic models of CO2 based on first-order perturbation theory. The first excited triplet state 3B2 of CO2, including its vibrational structure, is considered in our model, and a more consistent approach to CO2 dissociation is also proposed. The model is benchmarked against a few academic zero-dimensional (0D) cases and compared to decomposition time measurements in a shock tube. Our model is shown to have reasonable predictive capabilities, and the CO2 + O ↔ CO + O2 reaction is found to have a key influence on the dissociation dynamics of CO2 shocked flows, warranting further theoretical studies. We conclude this study with a discussion on the theoretical improvements that are still required for a more consistent analysis of the vibrational/dissociation dynamics of CO2.

Damped harmonic oscillator and bosonic string theory through noncommutativity

Some features of noncommutativity are investigated in this work. A noncommutative (NC) versions for the 2D harmonic oscillator and scalar field theory are obtained via the Noncommutative Mapping and fourfold discussion are presented: first, the Lagrangian of the damped harmonic oscillator (DHO), which was an original explanation for the dissipative process presenting in the atomic nuclei collision, is reobtained through noncommutativity. Here, we show that the NC parameter plays the role of the viscous damping coefficient; second, the NC harmonic oscillator is mapped into the bosonic string theory in the light cone frame, which appears as a bosonic string theory attached to a D3-brane; third, we discuss the connection between DHO and bosonic string attached to a D3-brane; fourth, the NC scalar field is indicating into 2D-NC harmonic oscillator.

Classical and Mexican hat-type potential energy curve for the hydrogen molecule from a confined Kratzer oscillator

We review harmonic oscillator theory for closed, stable quantum systems. The H2 potential energy curve (PEC) of Mexican hat-type, calculated with a confined Kratzer oscillator, is better than the Rydberg–Klein–Rees (RKR) H2 PEC. Compared with QM, the theory of chemical bonding is simplified, since a confined Kratzer oscillator can also lead to the long sought for universal function, once called the Holy Grail of Molecular Spectroscopy. This is validated by reducing PECs for different bonds H2, HF, I2, N2 and O2 to a single one. The equal probability for H2, originating either from [Formula: see text] or [Formula: see text], is quantified with a Gauss probability density function. At the Bohr scale, a confined harmonic oscillator behaves properly at the extremes of bound two-nucleon quantum systems.

Functional determinant of the massive Laplace operator and the multiplicative anomaly

After a brief survey of zeta function regularization issues and of the related multiplicative anomaly, illustrated with a couple of basic examples, namely the harmonic oscillator and quantum field theory at finite temperature, an application of these methods to the computation of functional determinants corresponding to massive Laplacians on spheres in arbitrary dimensions is presented. Explicit formulas are provided for the Laplace operator on spheres in dimensions and for ‘vector’ and ‘tensor’ Laplacians on the unitary sphere S4.

Influence of squeeze-film damping on higher-mode microcantilever vibrations in liquid

The functionality of atomic force microscopy (AFM) and nanomechanical sensing can be enhanced using higher-mode microcantilever vibrations. Both methods require a resonating microcantilever to be placed close to a surface, either a sample or the boundary of a microfluidic channel. Below a certain cantilever-surface separation, the confined fluid induces squeeze-film damping. Since damping changes the dynamic properties of the cantilever and decreases its sensitivity, it should be considered and minimized. Although squeeze-film damping in gases is comprehensively described, little experimental data is available in liquids, especially for higher-mode vibrations. We have measured the flexural higher-mode response of photothermally driven microcantilevers vibrating in water, close to a parallel surface with gaps ranging from ~200 μm to ~1 μm. A modified model based on harmonic oscillator theory was used to determine the modal eigenfrequencies and quality factors, which can be converted into co-moving fluid mass and dissipation coefficients. The range of squeeze-film damping between the cantilever and surface decreased for eigenfrequencies (inertial forces) and increased for quality factors (dissipative forces) with higher mode number. The results can be employed to improve the quantitative analysis of AFM measurements, design miniaturized sensor fluid cells, or benchmark theoretical models. PACS: 07.10.Cm (Micromechanical devices and systems), 46.40.Ff (Resonance and damping of mechanical waves), 07.79.-v (Scanning probe microscopes and components), 07.07.Df (Sensors (chemical, optical, electrical, movement, gas, etc.); remote sensing).

Topography-Correlated Confocal Raman Microscopy with Cylindrical Vector Beams for Probing Nanoscale Structural Order.

Cylindrical vector beams, such as radially or azimuthally polarized doughnut beams, are combined with topography studies of pentacene thin films, allowing us to correlate Raman spectroscopy with intermolecular interactions depending on the particular pentacene polymorph. Polarization-dependent Raman spectra of the C-H bending vibrations are resolved layer by layer within a thin film of ∼20 nm thickness. The variation of the Raman peak positions indicates changes in the molecular orientation and in the local environment at different heights of the pentacene film. With the assistance of a theoretical model based on harmonic oscillator and perturbation theory, our method reveals the local structural order and the polymorph at different locations within the same pentacene thin film, depending mainly on its thickness. In good agreement with the crystallographic structures reported in the literature, our observations demonstrate that the first few monolayers grown in a structure are closer to the thin-film phase, but for larger film thicknesses, the morphology evolves toward the crystal-bulk phase with a larger tilting angle of the pentacene molecules against the substrate normal.

Floquet Eigenvectors Theory of Pulsed Bias Phase and Quadrature Harmonic Oscillators

The paper presents an analytical derivation of Floquet eigenvalues and eigenvectors for a class of harmonic phase and quadrature oscillators. The derivation refers in particular to systems modeled by two parallel RLC resonators with pulsed energy restoring. Pulsed energy restoring is obtained through parallel current generators with an impulsive characteristic triggered by the resonators voltages. In performing calculation the initial hypothesis of the existence of stable oscillation is only made, then it is verified when both oscillation amplitude and eigenvalues/eigenvectors are deduced from symmetry conditions on oscillator space state. A detailed determination of the first eigenvector is obtained. Remaining eigenvectors are hence calculated with realistic approximations. Since Floquet eigenvectors are acknowledged to give the correct decomposition of noise perturbations superimposed to the oscillator space state along its limit cycle, an analytical and compact model of their behavior highlights the unique phase noise properties of this class of oscillators.

State-specific dissociation modeling with multiquantum vibration-translation transitions

An efficient state-specific model of dissociation using a quadrature approach is proposed by considering multiquantum vibration-translation (VT) transitions (between bound and quasibound vibrational levels) according to the forced harmonic oscillator theory. Application of this model to a pure N₂ gas flow behind a plane shock wave shows that dissociation has a large influence on intermediate and higher vibrational levels and is characterized by state-specific incubation distances, before which VT energy exchanges remain the dominant mechanism just behind the shock.

Simple model for vibration-translation exchange at high temperatures: Effects of multiquantum transitions on the relaxation of aN2gas flow behind a shock

In this paper a simple model is proposed for computation of rate coefficients related to vibration-translation transitions based on the forced harmonic oscillator theory. This model, which is developed by considering a quadrature method, provides rate coefficients that are in very good agreement with those found in the literature for the high temperature regime (≳10,000 K). This model is implemented to study a one-dimensional nonequilibrium inviscid N(2) flow behind a plane shock by considering a state-to-state approach. While the effects of ionization and chemical reactions are neglected in our study, our results show that multiquantum transitions have a great influence on the relaxation of the macroscopic parameters of the gas flow behind the shock, especially on vibrational distributions of high levels. All vibrational states are influenced by multiquantum processes, but the effective number of transitions decreases inversely according to the vibrational quantum number. For the initial conditions considered in this study, excited electronic states are found to be weakly populated and can be neglected in modeling. Moreover, the computing time is considerably reduced with the model described in this paper compared to others found in the literature.

Dynamical transition in proteins and non-Gaussian behavior of low-frequency modes in self-consistent normal mode analysis

Self-consistent normal mode analysis (SCNMA) is applied to heme c type cytochrome f to study temperature-dependent protein motion. Classical normal mode analysis assumes harmonic behavior and the protein mean-square displacement has a linear dependence on temperature. This is only consistent with low-temperature experimental results. To connect the protein vibrational motions between low and physiological temperatures, we have incorporated a fitted set of anharmonic potentials into SCNMA. In addition, quantum harmonic-oscillator theory has been used to calculate the displacement distribution for individual vibrational modes. We find that the modes involving soft bonds exhibit significant non-Gaussian dynamics at physiological temperature, which suggests that it may be the cause of the non-Gaussian behavior of the protein motions probed by elastic incoherent neutron scattering. The combined theory displays a dynamical transition caused by the softening of few "torsional" modes in the low-frequency regime ( <50 cm(-1) or <6 meV or >0.6 ps). These modes change from Gaussian to a classical distribution upon heating. Our theory provides an alternative way to understand the microscopic origin of the protein dynamical transition.

Squeezing and broadening effects in mechanical oscillators

The general theory of harmonic oscillators with time-dependent frequency and ‘effective mass’ is related in the present work to the use of the adiabatic approximation. A slow ‘pumping’ of energy into the mechanical oscillator leads to a slow increase of the amplitude and frequency with squeezing effects by which the noise in one quadrature is increasing and in the other quadrature is decreasing. An order of magnitude for the squeezing effect is calculated. It is suggested to use this theory for mechanical oscillators in traps and for resonant detectors. The other extreme case of ‘impulse’ interaction in which the time of interaction is very small relative to the time period of the harmonic oscillator is analyzed. In the ‘impact’ approximation the time development is without any squeezing, but broadening effects might be important under special conditions. The ‘impact’ approximation might be related to the detection of gravitational waves by Michelson interferometers.

Nonequilibrium Dissociation Processes in Hyperbolic Atmospheric Entries

Physical-chemical processes encountered behind atmospheric entry shock waves are known to occur in extremely nonequilibrium conditions. Also, translational temperatures up to 100,000 K may be reached immediately behind the shock wave. Such challenging conditions require the development of adequate state-to-state models, which prevents using widespread first-order theories. A complete database for the simulation of state-resolved dissociation processes is presented in this paper. Rate coefficients valid up to very high temperatures have been obtained for diatom-diatom collisions, using the forced harmonic oscillator theory. The rate coefficients for atom-diatom collisions have been selected after a critical review of the existing data sets, as the forced harmonic oscillator theory proved inadequate for the simulation of such processes. Such a consistent state-to-state model has then been used for simulating nitrogen dissociation processes behind very high-temperature shock waves, and the obtained results have been compared with those obtained using popular one- and two-temperature models.

GENERALIZED HEISENBERG RELATION AND QUANTUM HARMONIC OSCILLATORS

We study the consequences of the generalized Heisenberg uncertainty relation which admits a minimal uncertainty in length such as the case in a theory of quantum gravity. In particular, the theory of quantum harmonic oscillators arising from such a generalized uncertainty relation is examined. We demonstrate that all the standard properties of the quantum harmonic oscillators prevail when we employ a generalized momentum. We also show that quantum electrodynamics and coherent photon states can be described in the familiar standard manner despite the generalized uncertainty principle.

Breathing-mode resonance of a complex plasma disk

We have experimentally characterized the breathing mode oscillation of a strongly-coupled, dusty plasma disk. Steady-state oscillations are excited by sinusoidally modulating the plasma density, creating a single-frequency, in-plane driving force. Resonance curves agree well with damped harmonic oscillator theory. A response at the second harmonic is observed and found to increase with the square of the driving force, indicating a quadratic nonlinearity.

Stark Broadening and Stimulated Raman Pumping in High-Resolution N Coherent Anti-Stokes Raman Scattering Spectra

In previous studies, the high-resolution N 2 coherent anti-Stokes Raman scattering (CARS) technique has been used to acquire pressure, temperature, and density measurements in high-speed supersonic e ows. In these lowdensity e ows, a tradeoffexistsbetween elevating theCARS signal strength with increasing pump- and Stokes-laser intensities and introducing Stark broadening and stimulated Raman pumping effects into the high-resolution N 2 CARS spectra. To explore these laser-induced perturbations, theCARS technique is used to acquire v =0 ! 1 and v =1 ! 2 CARS spectra over a range of pressures in an optically accessible gas cell. By controlling the intensity of the pump- and Stokes-laser beams, Stark broadening effects in the high-resolution (D ! =0:10 cm i 1 ) broadband CARS spectra are explored. For pump-laser intensities greater than 185 GW/cm 2 , the least-squares e ts of the experimental spectra with theoretical spectra provide pressures and temperatures that diverge from conditions measuredwithin thecell usingconventional transducersforpressuresandtemperaturesaround0.2 atm and298K, respectively. An analytical model based on rigid-rotator harmonic-oscillator theory is used to describe how the increased optical e elds of the pump and Stokes lasers stretch the molecular bond between the nitrogen nuclei, broadening and shifting the rotational transitions in the Q-branch manifold. Finally, by increasing the pump- and Stokes-laser intensities simultaneously, ambient v=1 ! 2 N2 CARS spectra, resulting from stimulated Raman pumpingeffects,areacquiredwithhighresolution.Least-squarese tsoftheseexperimentalspectrawiththeoretical spectra show that stimulated Raman pumping signie cantly increases the vibrational temperature extracted from the experimental spectra. The relative intensities of the rotational transitions in the v=0 ! 1 manifold, however, are not affected by the stimulated Raman pumping process.

A new method to measure the oscillation of a cylindrical cantilever: “The laser reflection detection system”

A new optical detection system for measuring the oscillation of cylindrical cantilevers has been designed. A laser beam is directed perpendicularly to the oscillating plane and is focused on the curved surface of the vibrating probe. The surface reflects the light and a second lens refocuses it onto a two-segment photodiode. The sensitivity of this method lies in the fact that a small displacement of the probe produces a large angular deflection of the reflected laser. Applications of this new system are presented in order to demonstrate its reliability, accuracy, sensitivity, and the possible use in a shear force microscope. All the results are finally analyzed by modeling the motion of the cantilevers using harmonic oscillator theory and the continuous model for oscillating bars. The agreement between experimental data and models is well inside the experimental errors confirming the possibility of using this system to accurately study the dynamics of cylindrical cantilevers.